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Mirrors > Home > MPE Home > Th. List > f1orn | Structured version Visualization version GIF version |
Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
f1orn | β’ (πΉ:π΄β1-1-ontoβran πΉ β (πΉ Fn π΄ β§ Fun β‘πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o2 6839 | . 2 β’ (πΉ:π΄β1-1-ontoβran πΉ β (πΉ Fn π΄ β§ Fun β‘πΉ β§ ran πΉ = ran πΉ)) | |
2 | eqid 2733 | . . 3 β’ ran πΉ = ran πΉ | |
3 | df-3an 1090 | . . 3 β’ ((πΉ Fn π΄ β§ Fun β‘πΉ β§ ran πΉ = ran πΉ) β ((πΉ Fn π΄ β§ Fun β‘πΉ) β§ ran πΉ = ran πΉ)) | |
4 | 2, 3 | mpbiran2 709 | . 2 β’ ((πΉ Fn π΄ β§ Fun β‘πΉ β§ ran πΉ = ran πΉ) β (πΉ Fn π΄ β§ Fun β‘πΉ)) |
5 | 1, 4 | bitri 275 | 1 β’ (πΉ:π΄β1-1-ontoβran πΉ β (πΉ Fn π΄ β§ Fun β‘πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β‘ccnv 5676 ran crn 5678 Fun wfun 6538 Fn wfn 6539 β1-1-ontoβwf1o 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3956 df-ss 3966 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 |
This theorem is referenced by: f1f1orn 6845 infdifsn 9652 efopnlem2 26165 cycpmcl 32275 |
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